Compound inequalities
4x+7<11 or 1-x<=-2
4x+7 < 11
4x < 4
x < 1
1-x <= -2
-x <= -3
x >= 3
so, x<1 or x>=3
see at
http://www.wolframalpha.com/input/?i=solve+4x%2B7%3C11+or+1-x%3C%3D-2
-8 < 10 + 2k <=2
To solve compound inequalities, you need to solve each inequality separately and then combine the results.
Let's start with the first inequality: 4x + 7 < 11
1. Subtract 7 from both sides to isolate the term with x:
4x + 7 - 7 < 11 - 7
4x < 4
2. Divide both sides by 4 to solve for x:
(4x) / 4 < 4 / 4
x < 1
Now let's move on to the second inequality: 1 - x <= -2
1. Add x to both sides to isolate the term with x:
1 - x + x <= -2 + x
1 <= -2 + x
2. Subtract 1 from both sides to solve for x:
1 - 1 <= -2 + x - 1
0 <= -1 + x
-1 + x >= 0
Thus, we have two separate inequalities: x < 1 and -1 + x >= 0.
To combine these two inequalities, we need to find the overlapping region between the solutions.
For x < 1, x can take any value less than 1, but not equal to 1.
For -1 + x >= 0, x can take any value greater than or equal to 1.
To express this as a combined inequality, we can use the "and" operator (∧) to combine both inequalities:
x < 1 ∧ x ≥ -1
This means that x can take any value that is less than 1 and greater than or equal to -1.
In interval notation, this can be written as (-∞, 1) ∪ [-1, ∞).